For the simulations of the main text, we assume that \(N_{\phi }=1 \,\) . For more details on variational ansätze for DNLS consult [1]}, [2]}. From the solution of Eqs. (REF ) and (REF ) we then obtain the boson field, \(\phi ^{0}_{\mathbf {\mathrm {i}}}\) , which is then used to obtain the impurity density \(u^{0}_{\mathbf {\mathrm {i}}}=g\bar{\phi }^{0}_{\mathbf {\mathrm {i}}}\phi _{\mathbf {\mathrm {i}}} \, \) . The impurity density then acts as an effective attractive potential for bosons by virtue of Eq. (REF ). By solving the associated eigenvalue problem, we obtain the Bose polaron as the bound state of the impurity potential, as depicted in Fig. (4) of the main text.
